Exporting Texture Files as PNG or JPG
First thing - you can export textures from blender without a problem using uv/image editor:


Just select the UV/Image editor as show in (1). Then select texture you want to save (2) and finally save as (3).

Second important thing, from what I understood you are trying to export materials including nodes from Blender to Unreal Engine? This won't work as you suspect it would.

MTL files exported together with OBJs have very little material information and are pretty much useless, not to mention the format is old and there seem to be better alternatives available nowadays.

Basically no 3d software allows you to freely move its own materials to other 3d engines, everyone has its own material making system, renderer etc and I don't think I have ever seen any 3d modeling software being compatibile with 3d engine in that field. 

What you should do, is export all the textures, including baked ones (by "baked" I mean textures with shadow/colour/light/normal informations baked into them) and then recreate the material inside the engine using those textures.